 Solution

ID: past-exam-of-the-soft-condensed-matter-course-of-the-university-of-cambridge/2012/paper-1/3/c/solution

Overlap means roughly one coil per coil volume. With segment volume

a^{3}

ϕ^{*} \sim \frac{N a ^{3}}{R _{g}^{3}} .

For a strong good solvent,

R_{g} \sim a N^{3/5}

, and hence

ϕ^{*} \sim N^{- 4/5} = (1 0^{4})^{- 4/5} ≃ 6.3 \times 1 0^{- 4},

or about 0.063% by volume in the unit-prefactor scaling convention. The segment length cancels. A coil has no sharply defined boundary, so using

4 π R_{g}^{3} /3

or retaining a size amplitude changes an order-one prefactor. This calculation is a scaling estimate, not an exact volume-fraction measurement.

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